Why adding transparent shaders don't result in transparency?
This all comes down to what the Transparent shader and the Add shader actually do. Think about it another way and the behaviour you are seeing makes more sense.

Instinctively, the Transparent shader is nothing, however, it is not. The shader defines what happens when a ray interacts with the surface - in the case of Transparent, the incoming ray hits the surface and the result is that the resultant ray leaves the surface travelling in the same direction and, if the surface is 100% white, is the same intensity as the incoming ray - ie, no change.

The Add shader takes multiple shaders and combines the result. So, adding two Transparent shaders we start with the incoming ray and each Transparent shader produces its own (unaffected) resultant ray. The Add shader combines these two rays and the result is a combined ray with twice the intensity of the original - ie, it actually amplifies the light.

If you look at your result you’ll see that this corresponds with what you see - where the environment light passes through a surface it becomes brighter, and the more surfaces it passes though the brighter ot becomes.